Proof.    
This is a classical fact: since the Grassmannian is formally defined as a quotient of the Stiefel manifold \(\mathbb {S}t(M,{\mathbb {C}}^{T})\)  (i.e. the set of \(T\times M\)  complex matrices \({\bf X}\)  such that \({\bf X}^H{\bf X}={\bf I}_M\) ) by the unitary group \(\mathcal {U}_M\) , the volume of \(\mathbb {G}r(M,{\mathbb {C}}^{T})\)  is the quotient of the volumes of the Stiefel and unitary matrices which is well–known, see for example [1]} (note that there exist several normalizations for the Riemannian structure of the classical groups, leading to different volume formulas. We use the standard that considers \(\mathbb {S}t(M,{\mathbb {C}}^{T})\)  and \(\mathcal {U}_M\)  as submanifolds of their ambient affine spaces, with the inherited structure).
       
